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This question already has an answer here:

When I encounter these two, I usually don't have a problem understanding their meaning in the given context. Still, I don't really know what makes them different from each other xD
-なくて still is relatively easy for me to determine, since its the negated て form.

その人がすきじゃなくて、会うに行く。
"Not liking that person, I go meet him/her."

I can't really determine what ないで is though, and therefore the difference between the two isn't completely clear to me.

留学するつもりだったんですが、母に行かないでほしいって言われたんです。
"I wanted to study abroad, but I was told by my mother that she wishes that I don't go."

So I do see that it is something along the lines of negation, but I still can't thoroughly classify it's exact form and function :D

marked as duplicate by istrasci, macraf, Dono, Chocolate, Blavius Jun 6 '17 at 5:00

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Technically, ないで is a way of saying don't, or without doing. You can look up more about it here.

ない has some behaviors that are similar to those of ~い adjectives. ~なくて is akin to putting a comma at the end of a complete thought that ends with ない and adding additional information.

この仕事{しごと}は簡単{かんたん}じゃない。つらいです。

This sentance can be rephrased using the なくて grammar:

この仕事は簡単じゃなくて、つらいです。

~ないで has a different nuance to it in that it is more of a command. My favorite translation of this is: "don't", or where that doesn't fit "having not done ___".

Your example sentence is a great example of the "don't" meaning:

留学するつもりだったんですが、母に行かないでほしいって言われたんです。

I was planning on doing study abroad, but mom says she doesn't want me to go.

Here is an example sentence using the "having not done___"

だめだけど、食{た}べないで仕事に行{い}くつもりです。

It's not a great idea, but I plan on going to work without eating first.

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